Ribbon Graphs, Quadratic Differentials on Riemann Surfaces, and Algebraic Curves Defined over $\bar Q$
Mulase, Motohico ; Penkava, Michael
arXiv, 9811024 / Harvested from arXiv
Text on arXiv
It is well known that there is a bijective correspondence between metric ribbon graphs and compact Riemann surfaces with meromorphic Strebel differentials. In this article, it is proved that Grothendieck's correspondence between dessins d'enfants and Belyi morphisms is a special case of this correspondence. For a metric ribbon graph with edge length 1, an algebraic curve over $\bar Q$ and a Strebel differential on it is constructed. It is also shown that the critical trajectories of the measured foliation that is determined by the Strebel differential recover the original metric ribbon graph. Conversely, for every Belyi morphism, a unique Strebel differential is constructed such that the critical leaves of the measured foliation it determines form a metric ribbon graph of edge length 1, which coincides with the corresponding dessin d'enfant.
Publié le : 1998-11-25
Classification:  Mathematical Physics,  Mathematics - Algebraic Geometry,  Primary: 32G15, 57R20, 81Q30. Secondary: 14H15, 30E15, 30E20, 30F30 
@article{9811024,
     author = {Mulase, Motohico and Penkava, Michael},
     title = {Ribbon Graphs, Quadratic Differentials on Riemann Surfaces, and
  Algebraic Curves Defined over $\bar Q$},
     journal = {arXiv},
     volume = {1998},
     number = {0},
     year = {1998},
     language = {en},
     url = {http://dml.mathdoc.fr/item/9811024}
}

Mulase, Motohico; Penkava, Michael. Ribbon Graphs, Quadratic Differentials on Riemann Surfaces, and
  Algebraic Curves Defined over $\bar Q$. arXiv, Tome 1998 (1998) no. 0, . http://gdmltest.u-ga.fr/item/9811024/